And this commutes with \(x\otimes {\mathbb {1}}\) . Indeed, since \(x\in \mathcal {A}_{\mathrm {r}}(W_{\mathrm {R}}) = {\mathcal {A}}_0({\mathbb {R}}_+)\)  and \(x^{\prime }\in \mathcal {A}_{\mathrm {r}}(W_{\mathrm {R}})^{\prime } = {\mathcal {A}}_0({\mathbb {R}}_-)\) ,
it follows that \({\hbox{\rm Ad\,}}\Gamma \left(\overline{\varphi }\left(\frac{p_k}{P_1}\right)\right)(x) \in {\mathcal {A}}_0({\mathbb {R}}_+)\) 
for any \(p_k \ge 0\)  by the result of
Longo and Witten [1]} (see also the beginning of this Section),
and by the fact that the spectral support of \(E_1\)  is positive.
Then the equality naturally restricts to \(\mathcal {H}_{\mathrm {r}}\otimes \mathcal {H}_{\mathrm {r}}\) .
